Consider a closed path in the parameter space \((d_0,d_x,d_y,d_z,v_{0z},v_{zz})\)  of the general 1D Hamiltonian (REF ).
Let \(\alpha \in [0,2\pi )\sim S^1\)  be the variable parameterizing this closed path, defined on a unit circle \(S^1\) .
This one-parameter subfamily of 1D Hamiltonians \(\hat{H}(\hat{p}_z,\alpha )\)  is periodic in \(\alpha \) ,
and as such, it can be viewed as one effective 2D Hamiltonian, in the 2D space of \((p_z,\alpha )\) , which is effectively a cylinder.
It belongs to class A, topologically nontrivial in 2D, and represents an effective 2D quantum anomalous Hall (QAH) system
(here, in accord with the common definition, “anomalous” just means that real magnetic field is not involved).
This construction is also known as the 1D adiabatic charge pump [1]}, [2]}.
